Scaling Up for Flow in Porous Media, October 13-18, 2008, Dubrovnik ON NUMERICAL UPSCALING FOR STOKES AND ON NUMERICAL UPSCALING FOR STOKES AND STOKES- -BRINKMAN FLOWS BRINKMAN FLOWS STOKES Oleg Iliev , Z.Lakdawala, J.Willems, Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany V.Starikovicius , Vilnius Gediminas Technical University, Lithuania P.Popov , Inst. Scientific Computation, Texas A&M University, USA October 14, 2008
Content Content 1. Motivation and aims 2. Basic solver 3. Multiple scales. Subgrid approach 4. Computer simulations 5. Perspectives
1. Motivation and aims 1. Motivation and aims Motivation and aims Motivation and aims
CFD simulations for filtration CFD simulations for filtration Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: Main criteria determining the performance of a filter element: depend on: 1) 1) 1) Pressure drop 1) Pressure drop Pressure drop Pressure drop – – – – flow rate ratio; flow rate ratio; flow rate ratio; flow rate ratio; microscale ( e.g. fibrous geometry microscale microscale microscale 1) Dirt storage capacity; 1) Dirt storage capacity; 1) 1) Dirt storage capacity; Dirt storage capacity; local deposition of particles, etc ), and 1) Size of penetrating particles. 1) 1) 1) Size of penetrating particles. Size of penetrating particles. Size of penetrating particles. macroscale macroscale ( e.g., filter element macroscale macroscale geometry, pressure, velocity distribution, etc .)
Challenges to CFD simulations Challenges to CFD simulations � Multiple scales (particles, fibres, pleats, ribs, housing, � Multiple scales (particles, fibres, pleats, ribs, housing,… …); ); � Time � Time- -dependent performance; dependent performance; � Shortening the design time and Needs for new design ideas; � Shortening the design time and Needs for new design ideas; � Virtual filter element design; � Virtual filter element design; � Extensive computational time; � Extensive computational time; � Parameters measurement or calculation (permeability, deposition � Parameters measurement or calculation (permeability, deposition rate,..) rate,..) � Validation of the numerical simulation results; � Validation of the numerical simulation results; � … � …
Multiple scales in filtration Particles level Filter components Filter element Complete system Nano scale Micro scale Millimeter Centimeter Meter (Navier-)Stokes in pore space (Navier-)Stokes-Brinkmann in fluid and coupled with stochastic in porous media coupled with ODE for particles, …. concentration of particles Particles-Fiber Dirt loading of Pleats in cartrige Flow within interaction the filtering medium filters Filter element Filter installation
2. Basic solver 2. Basic solver Basic solver Basic solver
Basic CFD solver: SuFiS SuFiS Basic CFD solver: � Grids: Cartesian grid � Finite volume discretization on cell-centred collocated grid � Chorin projection method with implicit treatment of Darcy term � Proper treatment of discontinuous coefficients in pressure- correction equation � Subgrid approach incorporated � Specialized for filtration applications � Paralleization
Macro scale: Flow through fluid and porous regions � ��� ��� � Darcy � � � � ∂ u Momentum ɶ − 1 −∇⋅ µ ∇ + ρ ∇ + µ + ∇ = ( u ) ( u , ) u K u p f Equations ∂ t ����� ����� � − Navier Stokes � Continuity ∇⋅ = u 0 Equation K can be fixed, or can change due to loading of the filtering medium
3. Multiple scales. Subgrid Subgrid approach approach 3. Multiple scales. Multiple scales. Subgrid Subgrid approach approach Multiple scales.
3. Multiple scales. Subgrid Subgrid approach approach 3. Multiple scales. - State of the art (Stokes to Darcy; Darcy to Darcy; two State of the art (Stokes to Darcy; Darcy to Darcy; two- -level DD for level DD for - multiscale) multiscale ) - Microscale Microscale to to mesoscale mesoscale upscaling upscaling (Stokes to Darcy or to Brinkman (Stokes to Darcy or to Brinkman - - Mesoscale Mesoscale to to macroscale macroscale upscaling upscaling (Brinkman to Brinkman) (Brinkman to Brinkman) -
3. Multiple scales. Known: Upscaling Upscaling Stokes to Darcy Stokes to Darcy 3. Multiple scales. Known: − ∆ = ∇ ν − u p 1 − µ = ∇ K u p i i ∇ = u 0 ∇ = u 0 +boundary conditions: +boundary conditions: � periodic (Sanchez Palencia) � periodic (Sanchez Palencia) � const. velocity ( � const. velocity (Allaire Allaire) ) � engineering approach � engineering approach
3. Multiple scales. Known: Darcy to Darcy 3. Multiple scales. Known: Darcy to Darcy ɶ − 1 − µ = ∇ − K u p 1 − µ = ∇ K u p i ∇ = i u 0 ∇ = u 0 +boundary conditions: +boundary conditions: � periodic � periodic � linear � linear � presure � presure drop+oscilatory drop+oscilatory � presure � presure drop+Neumann drop+Neumann Note: Note: Some results available for Some results available for Macroheterogeneous case case Macroheterogeneous (block permeability, (block permeability, e.g., Wu, Efendiev,Hou Efendiev,Hou) ) e.g., Wu,
3. Multiple scales. Brinkman to Darcy or Brinkman 3. Multiple scales. Brinkman to Darcy or Brinkman
Multiple scales. Subgrid Subgrid approach approach Multiple scales. � Choose a basic grid on which the simulations are possible; � Choose a basic grid on which the simulations are possible; � Provide information about the fine geometrical details; � Provide information about the fine geometrical details; � For each grid cell check if it overlaps unresolved fine geometr � For each grid cell check if it overlaps unresolved fine geometry y details details � In marked cells (or their agglomeration) solve auxiliary proble � In marked cells (or their agglomeration) solve auxiliary problems ms on fine grid, and calculate effective permeability tensor; on fine grid, and calculate effective permeability tensor; � Solve the modified equations on the chosen grid (the fine detai � Solve the modified equations on the chosen grid (the fine details ls are accounted via the effective permeability). are accounted via the effective permeability). Example of selected location for which effective permeability is calculated
Multiple scales. Subgrid Subgrid approach approach Multiple scales. Usage of the subgrid subgrid approach: approach: Usage of the � Upscale and solve � Upscale and solve upscaled upscaled equations; equations; � Upscale, solve � Upscale, solve upscaled upscaled equations and prolong the solution to equations and prolong the solution to the fine scale; the fine scale; � Iterate over scales (two � Iterate over scales (two- -level DD with level DD with upscaling upscaling- -based coarse based coarse scale operator). scale operator). Open problems: Open problems: - No theory for No theory for upscaling upscaling blocks containing solid, blocks containing solid, - porous and fluid; porous and fluid; - No theory for No theory for macroheterogeneous macroheterogeneous case; case; - - … ….. .. -
4. Computer simulations 4. Computer simulations Computer simulations using Computer simulations using subgrid approach approach subgrid
4. Computer simulations 4. Computer simulations Pleated filter, simulations with Pleated filter, simulations with subgrid approach approach subgrid
4. Computer simulations 4. Computer simulations
Macro scale: Flow through fluid and porous regions
5. Perspectives 5. Perspectives Perspectives Perspectives
Multiscale Microscale Macroscale Particles motion Permeability and deposition Electrical filed Upscaling Upscaling Navier-Stokes-Brinkman Downscaling Downscaling Stokes Particles concentration Microstructures: www.geodict.com Filter Rate of elements deposition design filtration filtration (life time) (clogging)
Thank you Thank you www.itwm.fhg.de Fraunhofer ITWM www.dasmod.de Dependable Adaptive Systems and Mathematical Modeling, TU Kaiserslautern
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